# Formulas
## Derivatives

### Cartesian Coordinates
$$
dl = dx\hat{\textbf{x}} + dy\hat{\textbf{y}} + dz\hat{\textbf{z}}
$$ (dl)

$$d\tau = dxdydz$$ (dtau)

#### Gradient

$$
\nabla t = \frac{\partial t}{\partial x}\hat{\textbf{x}} +\frac{\partial t}{\partial y}\hat{\textbf{y}} + \frac{\partial t}{\partial z}\hat{\textbf{z}}
$$ (cart-grad)

#### Divergence

$$\nabla \cdot \textbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}$$ (cart-div)

#### Curl

$$\nabla \times \textbf{v} = (\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z})\hat{\textbf{x}} + (\frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x})\hat{\textbf{y}} + (\frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y})\hat{\textbf{z}}
$$ (cart-curl)

$$
\nabla \times \textbf{v} = \begin{vmatrix}
\hat{\textbf{x}} & \hat{\textbf{y}} & \hat{\textbf{z}} \\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\
v_x & v_y & v_z \\
\end{vmatrix}
$$ (cart-curl-mat)

#### Laplacian

$$
\nabla ^2 T = \nabla \cdot (\nabla T) = \frac{\partial^2T}{\partial x} + \frac{\partial^2T}{\partial y} + \frac{\partial^2T}{\partial z}
$$ (cart-laplacian)

### Spherical Coordinates

$$
dl = dr\hat{\textbf{r}} + rd\theta \hat{\bm{\theta}} + r \sin{\bm{\theta}} d\phi \hat{\bm{\phi}}
$$ (sphere-dl)

#### Gradient

#### Divergence

#### Curl

#### Laplacian


$$

$$
$$